19. The results of the last two chapters may be summed up as follows:--We have extended the conception of a series obtained in the first chapter; for we have found that these series are mostly presented to us in groups. These groups are found upon examination to be formed upon approximately the same type throughout a very wide and varied range of experience; the causes of this agreement we discussed and explained in some detail. When, however, we extend our examination by supposing the series to run to a very great length, we find that they may be divided into two classes separated by important distinctions. In one of these classes (that containing the results of games of chance) the conditions of production, and consequently the laws of statistical occurrence, may be practically regarded as absolutely fixed; and the extent of the divergences from the mean seem to know no finite limit. In the other class, on the contrary (containing the bulk of ordinary statistical enquiries), the conditions of production vary with more or less rapidity, and so in consequence do the results. Moreover it is often impossible that variations from the mean should exceed a certain amount. The former we may term _ideal_ series. It is they alone which show the requisite characteristics with any close approach to accuracy, and to make the theory of the subject tenable, we have really to substitute one of this kind for one of the less perfect ones of the other class, when these latter are under treatment. The former class have, however, been too exclusively considered by writers on the subject; and conceptions appropriate only to them, and not always even to them, have been imported into the other class. It is in this way that a general tendency to an excessive deductive or _à priori_ treatment of the science has been encouraged. 1. This latter enquiry belongs to what may be termed the more purely logical part of this volume, and is entered on in the course of Chapter VI. 2. For the use of those not acquainted with the common notation employed in this subject, it may be remarked that HH is simply an abbreviated way of saying that the two successive throws of the penny give head; HT that the first of them gives head, and the second tail; and so on with the remaining symbols. 3. I am endeavouring to treat this rule of Sufficient Reason in a way that shall be legitimate in the opinion of those who accept it, but there seem very great doubts whether a contradiction is not involved when we attempt to extract results from it. If the sides are absolutely alike, how can there he any difference between the terms of the series? The succession seems then reduced to a dull uniformity, a mere iteration of the same thing many times; the series we contemplated has disappeared. If the sides are not absolutely alike, what becomes of the applicability of the rule? 4. _Formal Logic_, p. 185. _Principles of Science_, p. 208.